I received the following answer after emailing a company that raises crickets.
We use a agricultural feed that includes feed beans, buckwheat, and feed corn non of which are edible by humans. We don't use any pre- or post-consumer waste to feed the crickets for two reasons: we can't control the consistency of taste with a non-regular feed stock and our food regulations won't allow us to feed them waste to then feed to humans. The companies using crickets and other insects as feed stock for fish and poultry are having a lot of success using pre- and post-consumer waste to feed their crickets.
This answer surprised me quite a bit. I've been eating beans, buckwheat, and corn for quite a while so I'm not sure why they're described as non-edible by humans. I thought maybe they were talking about some other part of the plant that humans don't eat, but they went on to say they don't use any pre-consumer waste.
So it seems like insect farms either need to grow/purchase agricultural food as feed for crickets, or they need to sell their crickets as feed for larger animals. Either way, that's a diversion of useful food energy that could have gone toward humans.
It seems like insect farming is less sustainable than growing plant-based food for humans.
Oct 2018 Update I just listened to an interview with a couple farmers from Caspian Acres near Kamloops, BC. At 6:10 the interview confirms that the black soldier fly larvae (BSFL) are fed entirely on food scraps from restaurants in the city, then the matured BSFL are used to feed ducks and other animals.